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I found this question and the other answer doesn't quite fit what I wanted. So I'm going to give my own answer. Note that this will only cover through season 5 of The Vampire Diaries and season 1 of The Originals. I may update it later, but the other answer suggests that this was when they had the most interaction.

If you want to know if you need to watch The Originals in order to watch The Vampire Diaries, you don't so far. I do think some events in The Originals benefit from seeing the second, third, fourth, and even fifth seasons of The Vampire Diaries.

In The Originals season 1, episode 1, events depicted in The Vampire Diaries season 4, episode 20, are shown from a different point of view.

In The Originals season 1, episodes 7 & 8, a character from The Vampire Diaries appears in The Originals. That character doesn't appear in episodes 7-10 of season 5 of The Vampire Diaries.

In The Vampire Diaries season 5, episode 11, three The Originals characters appear in scenes. The previous crossover story line is resolved.

In The Originals season 1, episode 20, an event from The Vampire Diaries causes side effects in The Originals.

It is therefore best (for seeing synergies) to watch The Vampire Diaries (TVD) 4x20, then The Originals (TO) 1x01 immediately after. I'd probably watch TVD 4x21 through 5x06 and then TO 1x02 to 1x06 (or vice versa; they're pretty independent). TO 1x07 and 1x08 after TVD 5x06. TVD 5x07 to 5x10 and TO 1x09 and TO 1x10 are pretty independent. It doesn't seem to make much difference that TVD 5x11 (the 100th episode) appeared after TO 1x11, but it did. Episodes TVD 5x12 to 5x18 and TO 1x12 to 1x18 seem independent. Starting around TO 1x19 and TVD 5x19, side effects from an event from The Vampire Diaries leak over to The Originals. If you want to see how they build in both shows, alternate between them.

It is likely that someone is reading this and saying that this is overly pedantic and obsessive compulsive. That it's not really necessary to do this, as events on the two shows don't really depend on each other. This is true. It's quite possible to watch the two shows separately and you won't be confused. But there are some connections between the shows. This is the viewing order that I wish I had known prior to watching the relevant episodes.

No, there's no real good reason to watch The Originals in any particular order to understand The Vampire Diaries, or vice versa. Unlike other shared universes (say, the Arrowverse), the shows are pretty compartmentalized.

Obviously, The Originals is the spin-off series, so you would definitely want to watch the first few seasons of The Vampire Diaries in order to figure out who the main characters are and what they're up to prior to Klaus heading over to New Orleans.

Seasons Two introduces Elijah and Klaus, explains a bit about the Original family, and why Klaus is special

Season Three introduces the rest of the Original family, including Mikael, and explains what happen to Kol and Finn.

Season Four introduces Hayley, and also includes the "backdoor pilot" for The Originals.

After that, The Originals remains largely separate from The Vampire Diaries. Season Five of the The Vampire Diaries also includes Kol as a recurring character, somewhat hinting at his future in The Originals, but nothing significant.

There are a handful of crossover episodes, but they don't have much of an impact on each other's storylines:

Tyler Lockwood shows up in New Orleans for two episodes, an event that is hinted at in The Vampire Diaries earlier in the season, but other than knowing that he dumped Caroline there's not much overlap.

Klaus and Rebekah return to Mystic Falls for the 100th episode of The Vampire Diaries but it's mostly to wrap up their respective storylines with Matt and Caroline -- nothing about New Orleans comes up.

Stefan visits New Orleans in Seasons 7/3 in a two-parter that's probably the most significant story crossover -- he's running from a vampire hunter -- but if you saw them out of context each episode is pretty understandable on its own.

Beyond that, there aren't even all that many callouts to the other shows. In fact, trying to merge the two continuities together often results in some nonsensical conclusions (neither show tends to even acknowledge the major changes in mythology that pop up every so often).

You can pretty much watch them in whatever order you want and you won't miss anything.